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Journal articles on the topic 'Forward Backward Stochastic Differential Equations (FBSDE)'

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Published: 1 February 2025

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1

Zhang, Kevin, Junhao Zhu, Dehan Kong, and Zhaolei Zhang. "Modeling single cell trajectory using forward-backward stochastic differential equations." PLOS Computational Biology 20, no. 4 (April 15, 2024): e1012015. http://dx.doi.org/10.1371/journal.pcbi.1012015.

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Recent advances in single-cell sequencing technology have provided opportunities for mathematical modeling of dynamic developmental processes at the single-cell level, such as inferring developmental trajectories. Optimal transport has emerged as a promising theoretical framework for this task by computing pairings between cells from different time points. However, optimal transport methods have limitations in capturing nonlinear trajectories, as they are static and can only infer linear paths between endpoints. In contrast, stochastic differential equations (SDEs) offer a dynamic and flexible approach that can model non-linear trajectories, including the shape of the path. Nevertheless, existing SDE methods often rely on numerical approximations that can lead to inaccurate inferences, deviating from true trajectories. To address this challenge, we propose a novel approach combining forward-backward stochastic differential equations (FBSDE) with a refined approximation procedure. Our FBSDE model integrates the forward and backward movements of two SDEs in time, aiming to capture the underlying dynamics of single-cell developmental trajectories. Through comprehensive benchmarking on multiple scRNA-seq datasets, we demonstrate the superior performance of FBSDE compared to other methods, highlighting its efficacy in accurately inferring developmental trajectories.

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2

Takahashi, Akihiko, and Toshihiro Yamada. "An asymptotic expansion of forward-backward SDEs with a perturbed driver." International Journal of Financial Engineering 02, no. 02 (June 2015): 1550020. http://dx.doi.org/10.1142/s2424786315500206.

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Motivated by nonlinear pricing in finance, this paper presents a mathematical validity of an asymptotic expansion scheme for a system of forward-backward stochastic differential equations (FBSDEs) in terms of a perturbed driver in the BSDE and a small diffusion in the FSDE. In particular, we represent the coefficients of the expansion of the FBSDE up to an arbitrary order, and obtain the error estimate of the expansion with respect to the driver and the small noise perturbation.

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3

Yang, Jie, and Weidong Zhao. "Convergence of Recent Multistep Schemes for a Forward-Backward Stochastic Differential Equation." East Asian Journal on Applied Mathematics 5, no. 4 (November 2015): 387–404. http://dx.doi.org/10.4208/eajam.280515.211015a.

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AbstractConvergence analysis is presented for recently proposed multistep schemes, when applied to a special type of forward-backward stochastic differential equations (FB-SDEs) that arises in finance and stochastic control. The corresponding k-step scheme admits a k-order convergence rate in time, when the exact solution of the forward stochastic differential equation (SDE) is given. Our analysis assumes that the terminal conditions and the FBSDE coefficients are sufficiently regular.

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4

Geiss, Christel, Céline Labart, and Antti Luoto. "Mean square rate of convergence for random walk approximation of forward-backward SDEs." Advances in Applied Probability 52, no. 3 (September 2020): 735–71. http://dx.doi.org/10.1017/apr.2020.17.

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AbstractLet (Y, Z) denote the solution to a forward-backward stochastic differential equation (FBSDE). If one constructs a random walk $B^n$ from the underlying Brownian motion B by Skorokhod embedding, one can show $L_2$-convergence of the corresponding solutions $(Y^n,Z^n)$ to $(Y, Z).$ We estimate the rate of convergence based on smoothness properties, especially for a terminal condition function in $C^{2,\alpha}$. The proof relies on an approximative representation of $Z^n$ and uses the concept of discretized Malliavin calculus. Moreover, we use growth and smoothness properties of the partial differential equation associated to the FBSDE, as well as of the finite difference equations associated to the approximating stochastic equations. We derive these properties by probabilistic methods.

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5

Ji, Shaolin, Chuanfeng Sun, and Qingmeng Wei. "The Dynamic Programming Method of Stochastic Differential Game for Functional Forward-Backward Stochastic System." Mathematical Problems in Engineering 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/958920.

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This paper is devoted to a stochastic differential game (SDG) of decoupled functional forward-backward stochastic differential equation (FBSDE). For our SDG, the associated upper and lower value functions of the SDG are defined through the solution of controlled functional backward stochastic differential equations (BSDEs). Applying the Girsanov transformation method introduced by Buckdahn and Li (2008), the upper and the lower value functions are shown to be deterministic. We also generalize the Hamilton-Jacobi-Bellman-Isaacs (HJBI) equations to the path-dependent ones. By establishing the dynamic programming principal (DPP), we derive that the upper and the lower value functions are the viscosity solutions of the corresponding upper and the lower path-dependent HJBI equations, respectively.

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6

Song, Yunquan. "Terminal-Dependent Statistical Inference for the FBSDEs Models." Mathematical Problems in Engineering 2014 (2014): 1–11. http://dx.doi.org/10.1155/2014/365240.

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The original stochastic differential equations (OSDEs) and forward-backward stochastic differential equations (FBSDEs) are often used to model complex dynamic process that arise in financial, ecological, and many other areas. The main difference between OSDEs and FBSDEs is that the latter is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. It is interesting but challenging to estimate FBSDE parameters from noisy data and the terminal condition. However, to the best of our knowledge, the terminal-dependent statistical inference for such a model has not been explored in the existing literature. We proposed a nonparametric terminal control variables estimation method to address this problem. The reason why we use the terminal control variables is that the newly proposed inference procedures inherit the terminal-dependent characteristic. Through this new proposed method, the estimators of the functional coefficients of the FBSDEs model are obtained. The asymptotic properties of the estimators are also discussed. Simulation studies show that the proposed method gives satisfying estimates for the FBSDE parameters from noisy data and the terminal condition. A simulation is performed to test the feasibility of our method.

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7

DOS REIS, GONÇALO, and RICARDO J. N. DOS REIS. "A NOTE ON COMONOTONICITY AND POSITIVITY OF THE CONTROL COMPONENTS OF DECOUPLED QUADRATIC FBSDE." Stochastics and Dynamics 13, no. 04 (October 7, 2013): 1350005. http://dx.doi.org/10.1142/s0219493713500056.

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In this note we are concerned with the solution of Forward–Backward Stochastic Differential Equations (FBSDE) with drivers that grow quadratically in the control component (quadratic growth FBSDE or qgFBSDE). The main theorem is a comparison result that allows comparing componentwise the signs of the control processes of two different qgFBSDE. As a by-product one obtains conditions that allow establishing the positivity of the control process.

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8

Wang, Mingcan, and Xiangjun Wang. "Hybrid Neural Networks for Solving Fully Coupled, High-Dimensional Forward–Backward Stochastic Differential Equations." Mathematics 12, no. 7 (April 3, 2024): 1081. http://dx.doi.org/10.3390/math12071081.

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The theory of forward–backward stochastic differential equations occupies an important position in stochastic analysis and practical applications. However, the numerical solution of forward–backward stochastic differential equations, especially for high-dimensional cases, has stagnated. The development of deep learning provides ideas for its high-dimensional solution. In this paper, our focus lies on the fully coupled forward–backward stochastic differential equation. We design a neural network structure tailored to the characteristics of the equation and develop a hybrid BiGRU model for solving it. We introduce the time dimension based on the sequence nature after discretizing the FBSDE. By considering the interactions between preceding and succeeding time steps, we construct the BiGRU hybrid model. This enables us to effectively capture both long- and short-term dependencies, thus mitigating issues such as gradient vanishing and explosion. Residual learning is introduced within the neural network at each time step; the structure of the loss function is adjusted according to the properties of the equation. The model established above can effectively solve fully coupled forward–backward stochastic differential equations, effectively avoiding the effects of dimensional catastrophe, gradient vanishing, and gradient explosion problems, with higher accuracy, stronger stability, and stronger model interpretability.

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9

Wu, Zhen. "Fully coupled FBSDE with Brownian motion and Poisson process in stopping time duration." Journal of the Australian Mathematical Society 74, no. 2 (April 2003): 249–66. http://dx.doi.org/10.1017/s1446788700003281.

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AbstractWe first give the existence and uniqueness result and a comparison theorem for backward stochastic differential equations with Brownian motion and Poisson process as the noise source in stopping time (unbounded) duration. Then we obtain the existence and uniqueness result for fully coupled forward-backward stochastic differential equation with Brownian motion and Poisson process in stopping time (unbounded) duration. We also proved a comparison theorem for this kind of equation.

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10

Wei, Qingmeng, Jiongmin Yong, and Zhiyong Yu. "Linear quadratic stochastic optimal control problems with operator coefficients: open-loop solutions." ESAIM: Control, Optimisation and Calculus of Variations 25 (2019): 17. http://dx.doi.org/10.1051/cocv/2018013.

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An optimal control problem is considered for linear stochastic differential equations with quadratic cost functional. The coefficients of the state equation and the weights in the cost functional are bounded operators on the spaces of square integrable random variables. The main motivation of our study is linear quadratic (LQ, for short) optimal control problems for mean-field stochastic differential equations. Open-loop solvability of the problem is characterized as the solvability of a system of linear coupled forward-backward stochastic differential equations (FBSDE, for short) with operator coefficients, together with a convexity condition for the cost functional. Under proper conditions, the well-posedness of such an FBSDE, which leads to the existence of an open-loop optimal control, is established. Finally, as applications of our main results, a general mean-field LQ control problem and a concrete mean-variance portfolio selection problem in the open-loop case are solved.

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11

Oyono Ngou, Polynice, and Cody Hyndman. "A Fourier Interpolation Method for Numerical Solution of FBSDEs: Global Convergence, Stability, and Higher Order Discretizations." Journal of Risk and Financial Management 15, no. 9 (August 31, 2022): 388. http://dx.doi.org/10.3390/jrfm15090388.

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The convolution method for the numerical solution of forward-backward stochastic differential equations (FBSDEs) was originally formulated using Euler time discretizations and a uniform space grid. In this paper, we utilize a tree-like spatial discretization that approximates the BSDE on the tree, so that no spatial interpolation procedure is necessary. In addition to suppressing extrapolation error, leading to a globally convergent numerical solution for the FBSDE, we provide explicit convergence rates. On this alternative grid the conditional expectations involved in the time discretization of the BSDE are computed using Fourier analysis and the fast Fourier transform (FFT) algorithm. The method is then extended to higher-order time discretizations of FBSDEs. Numerical results demonstrating convergence are presented using a commodity price model, incorporating seasonality, and forward prices.

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12

Sun, Jingrui, and Hanxiao Wang. "Linear-quadratic optimal control for backward stochastic differential equations with random coefficients." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 46. http://dx.doi.org/10.1051/cocv/2021049.

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This paper is concerned with a linear-quadratic (LQ, for short) optimal control problem for backward stochastic differential equations (BSDEs, for short), where the coefficients of the backward control system and the weighting matrices in the cost functional are allowed to be random. By a variational method, the optimality system, which is a coupled linear forward-backward stochastic differential equation (FBSDE, for short), is derived, and by a Hilbert space method, the unique solvability of the optimality system is obtained. In order to construct the optimal control, a new stochastic Riccati-type equation is introduced. It is proved that an adapted solution (possibly non-unique) to the Riccati equation exists and decouples the optimality system. With this solution, the optimal control is obtained in an explicit way.

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13

Tang, Maoning. "A Variational Formula for Nonzero-Sum Stochastic Differential Games of FBSDEs and Applications." Mathematical Problems in Engineering 2014 (2014): 1–9. http://dx.doi.org/10.1155/2014/283418.

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A nonzero-sum stochastic differential game problem is investigated for fully coupled forward-backward stochastic differential equations (FBSDEs in short) where the control domain is not necessarily convex. A variational formula for the cost functional in a given spike perturbation direction of control processes is derived by the Hamiltonian and associated adjoint systems. As an application, a global stochastic maximum principle of Pontryagin’s type for open-loop Nash equilibrium points is established. Finally, an example of a linear quadratic nonzero-sum game problem is presented to illustrate that the theories may have interesting practical applications and the corresponding Nash equilibrium point is characterized by the optimality system. Here the optimality system is a fully coupled FBSDE with double dimensions (DFBSDEs in short) which consists of the state equation, the adjoint equation, and the optimality conditions.

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14

Fu, Yu, and Weidong Zhao. "An Explicit Second-Order Numerical Scheme to Solve Decoupled Forward Backward Stochastic Equations." East Asian Journal on Applied Mathematics 4, no. 4 (November 2014): 368–85. http://dx.doi.org/10.4208/eajam.030614.171014a.

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AbstractAn explicit numerical scheme is proposed for solving decoupled forward backward stochastic differential equations (FBSDE) represented in integral equation form. A general error inequality is derived for this numerical scheme, which also implies its stability. Error estimates are given based on this inequality, showing that the explicit scheme can be second-order. Some numerical experiments are carried out to illustrate the high accuracy of the proposed scheme.

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15

Casserini, Matteo, and Gechun Liang. "Fully coupled forward–backward stochastic dynamics and functional differential systems." Stochastics and Dynamics 15, no. 02 (April 6, 2015): 1550006. http://dx.doi.org/10.1142/s0219493715500069.

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This paper introduces and solves a general class of fully coupled forward–backward stochastic dynamics by investigating the associated system of functional differential equations. As a consequence, we are able to solve many different types of forward–backward stochastic differential equations (FBSDEs) that do not fit in the classical setting. In our approach, the equations are running in the same time direction rather than in a forward and backward way, and the conflicting nature of the structure of FBSDEs is therefore avoided.

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16

Chen, Li, Zhen Wu, and Zhiyong Yu. "Delayed Stochastic Linear-Quadratic Control Problem and Related Applications." Journal of Applied Mathematics 2012 (2012): 1–22. http://dx.doi.org/10.1155/2012/835319.

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We discuss a quadratic criterion optimal control problem for stochastic linear system with delay in both state and control variables. This problem will lead to a kind of generalized forward-backward stochastic differential equations (FBSDEs) with Itô’s stochastic delay equations as forward equations and anticipated backward stochastic differential equations as backward equations. Especially, we present the optimal feedback regulator for the time delay system via a new type of Riccati equations and also apply to a population optimal control problem.

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17

Liu, Meijuan, Xiangrong Wang, and Hong Huang. "Maximum Principle for Forward-Backward Control System Driven by Itô-Lévy Processes under Initial-Terminal Constraints." Mathematical Problems in Engineering 2017 (2017): 1–13. http://dx.doi.org/10.1155/2017/1868560.

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This paper investigates a stochastic optimal control problem where the control system is driven by Itô-Lévy process. We prove the necessary condition about existence of optimal control for stochastic system by using traditional variational technique under the assumption that control domain is convex. We require that forward-backward stochastic differential equations (FBSDE) be fully coupled, and the control variable is allowed to enter both diffusion and jump coefficient. Moreover, we also require that the initial-terminal state be constrained. Finally, as an application to finance, we show an example of recursive consumption utility optimization problem to illustrate the practicability of our result.

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18

Min, Hui, Ying Peng, and Yongli Qin. "Fully Coupled Mean-Field Forward-Backward Stochastic Differential Equations and Stochastic Maximum Principle." Abstract and Applied Analysis 2014 (2014): 1–15. http://dx.doi.org/10.1155/2014/839467.

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We discuss a new type of fully coupled forward-backward stochastic differential equations (FBSDEs) whose coefficients depend on the states of the solution processes as well as their expected values, and we call them fully coupled mean-field forward-backward stochastic differential equations (mean-field FBSDEs). We first prove the existence and the uniqueness theorem of such mean-field FBSDEs under some certain monotonicity conditions and show the continuity property of the solutions with respect to the parameters. Then we discuss the stochastic optimal control problems of mean-field FBSDEs. The stochastic maximum principles are derived and the related mean-field linear quadratic optimal control problems are also discussed.

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19

Zhao, Weidong, Wei Zhang, and Lili Ju. "A Multistep Scheme for Decoupled Forward-Backward Stochastic Differential Equations." Numerical Mathematics: Theory, Methods and Applications 9, no. 2 (May 2016): 262–88. http://dx.doi.org/10.4208/nmtma.2016.m1421.

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AbstractUpon a set of backward orthogonal polynomials, we propose a novel multi-step numerical scheme for solving the decoupled forward-backward stochastic differential equations (FBSDEs). Under Lipschtiz conditions on the coefficients of the FBSDEs, we first get a general error estimate result which implies zero-stability of the proposed scheme, and then we further prove that the convergence rate of the scheme can be of high order for Markovian FBSDEs. Some numerical experiments are presented to demonstrate the accuracy of the proposed multi-step scheme and to numerically verify the theoretical results.

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20

Drapeau, Samuel, Peng Luo, Alexander Schied, and Dewen Xiong. "An FBSDE approach to market impact games with stochastic parameters." Probability, Uncertainty and Quantitative Risk 6, no. 3 (2021): 237. http://dx.doi.org/10.3934/puqr.2021012.

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<p style='text-indent:20px;'>In this study, we have analyzed a market impact game between <i>n</i> risk-averse agents who compete for liquidity in a market impact model with a permanent price impact and additional slippage. Most market parameters, including volatility and drift, are allowed to vary stochastically. Our first main result characterizes the Nash equilibrium in terms of a fully coupled system of forward-backward stochastic differential equations (FBSDEs). Our second main result provides conditions under which this system of FBSDEs has a unique solution, resulting in a unique Nash equilibrium. </p>

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21

Tang, Tao, Weidong Zhao, and Tao Zhou. "Deferred Correction Methods for Forward Backward Stochastic Differential Equations." Numerical Mathematics: Theory, Methods and Applications 10, no. 2 (May 2017): 222–42. http://dx.doi.org/10.4208/nmtma.2017.s02.

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AbstractThe deferred correction (DC) method is a classical method for solving ordinary differential equations; one of its key features is to iteratively use lower order numerical methods so that high-order numerical scheme can be obtained. The main advantage of the DC approach is its simplicity and robustness. In this paper, the DC idea will be adopted to solve forward backward stochastic differential equations (FBSDEs) which have practical importance in many applications. Noted that it is difficult to design high-order and relatively “clean” numerical schemes for FBSDEs due to the involvement of randomness and the coupling of the FSDEs and BSDEs. This paper will describe how to use the simplest Euler method in each DC step–leading to simple computational complexity–to achieve high order rate of convergence.

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22

Aazizi, Soufiane. "Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes." Advances in Applied Probability 45, no. 3 (September 2013): 791–821. http://dx.doi.org/10.1239/aap/1377868539.

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We present a new algorithm to discretize a decoupled forward‒backward stochastic differential equation driven by a pure jump Lévy process (FBSDEL for short). The method consists of two steps. In the first step we approximate the FBSDEL by a forward‒backward stochastic differential equation driven by a Brownian motion and Poisson process (FBSDEBP for short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps ε goes to 0. In the second step we obtain the Lp-Hölder continuity of the solution of the FBSDEBP and we construct two numerical schemes for this FBSDEBP. Based on the Lp-Hölder estimate, we prove the convergence of the scheme when the number of time steps n goes to ∞. Combining these two steps leads to the proof of the convergence of numerical schemes to the solution of FBSDEs driven by pure jump Lévy processes.

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23

Aazizi, Soufiane. "Discrete-Time Approximation of Decoupled Forward‒Backward Stochastic Differential Equations Driven by Pure Jump Lévy Processes." Advances in Applied Probability 45, no. 03 (September 2013): 791–821. http://dx.doi.org/10.1017/s0001867800006583.

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We present a new algorithm to discretize a decoupled forward‒backward stochastic differential equation driven by a pure jump Lévy process (FBSDEL for short). The method consists of two steps. In the first step we approximate the FBSDEL by a forward‒backward stochastic differential equation driven by a Brownian motion and Poisson process (FBSDEBP for short), in which we replace the small jumps by a Brownian motion. Then, we prove the convergence of the approximation when the size of small jumps ε goes to 0. In the second step we obtain theLp-Hölder continuity of the solution of the FBSDEBP and we construct two numerical schemes for this FBSDEBP. Based on theLp-Hölder estimate, we prove the convergence of the scheme when the number of time stepsngoes to ∞. Combining these two steps leads to the proof of the convergence of numerical schemes to the solution of FBSDEs driven by pure jump Lévy processes.

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24

Cruzeiro, Ana Bela, André de Oliveira Gomes, and Liangquan Zhang. "Asymptotic properties of coupled forward–backward stochastic differential equations." Stochastics and Dynamics 14, no. 03 (May 29, 2014): 1450004. http://dx.doi.org/10.1142/s021949371450004x.

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In this paper, we consider coupled forward–backward stochastic differential equations (FBSDEs in short) with parameter ε > 0, of the following type [Formula: see text] We study the asymptotic behavior of its solutions and establish a large deviation principle for the corresponding processes.

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25

Zhang, Wei, and Hui Min. "Weak Convergence Analysis and Improved Error Estimates for Decoupled Forward-Backward Stochastic Differential Equations." Mathematics 9, no. 8 (April 13, 2021): 848. http://dx.doi.org/10.3390/math9080848.

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In this paper, we mainly investigate the weak convergence analysis about the error terms which are determined by the discretization for solving the stochastic differential equation (SDE, for short) in forward-backward stochastic differential equations (FBSDEs, for short), which is on the basis of Itô Taylor expansion, the numerical SDE theory, and numerical FBSDEs theory. Under the weak convergence analysis of FBSDEs, we further establish better error estimates of recent numerical schemes for solving FBSDEs.

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Ma, Jin, and Jakša Cvitanić. "Reflected forward-backward SDEs and obstacle problems with boundary conditions." Journal of Applied Mathematics and Stochastic Analysis 14, no. 2 (January 1, 2001): 113–38. http://dx.doi.org/10.1155/s1048953301000090.

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In this paper we study a class of forward-backward stochastic differential equations with reflecting boundary conditions (FBSDER for short). More precisely, we consider the case in which the forward component of the FBSDER is restricted to a fixed, convex region, and the backward component will stay, at each fixed time, in a convex region that may depend on time and is possibly random. The solvability of such FBSDER is studied in a fairly general way. We also prove that if the coefficients are all deterministic and the backward equation is one-dimensional, then the adapted solution of such FBSDER will give the viscosity solution of a quasilinear variational inequality (obstacle problem) with a Neumann boundary condition. As an application, we study how the solvability of FBSDERs is related to the solvability of an American game option.

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27

Wang, Yanqing, and Zhiyong Yu. "On the partial controllability of SDEs and the exact controllability of FBSDES." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 68. http://dx.doi.org/10.1051/cocv/2019052.

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A notion of partial controllability (also can be called directional controllability or output controllability) is proposed for linear controlled (forward) stochastic differential equations (SDEs), which characterizes the ability of the state to reach some given random hyperplane. It generalizes the classical notion of exact controllability. For time-invariant system, checkable rank conditions ensuring SDEs’ partial controllability are provided. With some special setting, the partial controllability for SDEs is proved to be equivalent to the exact controllability for linear controlled forward-backward stochastic differential equations (FBSDEs). Moreover, we obtain some equivalent conclusions to partial controllability for SDEs or exact controllability for FBSDEs, including the validity of observability inequalities for the adjoint equations, the solvability of some optimal control problems, the solvability of norm optimal control problems, and the non-singularity of a random version of Gramian matrix.

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Gong, Bo, and Weidong Zhao. "Optimal Error Estimates for a Fully Discrete Euler Scheme for Decoupled Forward Backward Stochastic Differential Equations." East Asian Journal on Applied Mathematics 7, no. 3 (August 2017): 548–65. http://dx.doi.org/10.4208/eajam.110417.070517a.

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AbstractIn error estimates of various numerical approaches for solving decoupled forward backward stochastic differential equations (FBSDEs), the rate of convergence for one variable is usually less than for the other. Under slightly strengthened smoothness assumptions, we show that the fully discrete Euler scheme admits a first-order rate of convergence for both variables.

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29

Zeng, Xiaoxiao, Kexin Fu, Xiaofei Li, Junjie Du, and Weiran Fan. "Numerical Method for Multi-Dimensional Coupled Forward-Backward Stochastic Differential Equations Based on Fractional Fourier Fast Transform." Fractal and Fractional 7, no. 6 (May 30, 2023): 441. http://dx.doi.org/10.3390/fractalfract7060441.

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Forward-backward stochastic differential equations (FBSDEs) have received more and more attention in the past two decades. FBSDEs can be applied to many fields, such as economics and finance, engineering control, population dynamics analysis, and so on. In most cases, FBSDEs are nonlinear and high-dimensional and cannot be obtained as analytic solutions. Therefore, it is necessary and important to design their numerical approximation methods. In this paper, a novel numerical method of multi-dimensional coupled FBSDEs is proposed based on a fractional Fourier fast transform (FrFFT) algorithm, which is used to compute the Fourier and inverse Fourier transforms. For the forward component of FBSDEs, time discretization is used as well as the backward equation to yield a recursive system with terminal conditions. For the numerical experiments to be successful, three types of numerical methods were used to solve the problem, which ensured the efficiency and speed of computation. Finally, the numerical methods for different examples are verified.

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30

Tang, Maoning. "Stochastic Maximum Principle of Near-Optimal Control of Fully Coupled Forward-Backward Stochastic Differential Equation." Abstract and Applied Analysis 2014 (2014): 1–12. http://dx.doi.org/10.1155/2014/361259.

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This paper first makes an attempt to investigate the near-optimal control of systems governed by fully nonlinear coupled forward-backward stochastic differential equations (FBSDEs) under the assumption of a convex control domain. By Ekeland’s variational principle and some basic estimates for state processes and adjoint processes, we establish the necessary conditions for anyε-near optimal control in a local form with an error order of exactε1/2. Moreover, under additional convexity conditions on Hamiltonian function, we prove that anε-maximum condition in terms of the Hamiltonian in the integral form is sufficient for near-optimality of orderε1/2.

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31

Li, Ruijing, and Chaozhu Hu. "Maximum Principle for Near-Optimality of Mean-Field FBSDEs." Mathematical Problems in Engineering 2020 (June 8, 2020): 1–16. http://dx.doi.org/10.1155/2020/8572959.

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The present paper concerns with a near-optimal control problem for systems governed by mean-field forward-backward stochastic differential equations (FBSDEs) with mixed initial-terminal conditions. Utilizing Ekeland’s variational principle as well as the reduction method, the necessary and sufficient near-optimality conditions are established in the form of Pontryagin’s type. The results are obtained under restriction on the convexity of the control domain. As an application, a linear-quadratic stochastic control problem is solved explicitly.

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32

Buckdahn, Rainer, and Ying Hu. "Hedging contingent claims for a large investor in an incomplete market." Advances in Applied Probability 30, no. 1 (March 1998): 239–55. http://dx.doi.org/10.1239/aap/1035228002.

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In this paper we study the problem of pricing contingent claims for a large investor (i.e. the coefficients of the price equation can also depend on the wealth process of the hedger) in an incomplete market where the portfolios are constrained. We formulate this problem so as to find the minimal solution of forward-backward stochastic differential equations (FBSDEs) with constraints. We use the penalization method to construct a sequence of FBSDEs without constraints, and we show that the solutions of these equations converge to the minimal solution we are interested in.

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Buckdahn, Rainer, and Ying Hu. "Hedging contingent claims for a large investor in an incomplete market." Advances in Applied Probability 30, no. 01 (March 1998): 239–55. http://dx.doi.org/10.1017/s0001867800008181.

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In this paper we study the problem of pricing contingent claims for a large investor (i.e. the coefficients of the price equation can also depend on the wealth process of the hedger) in an incomplete market where the portfolios are constrained. We formulate this problem so as to find the minimal solution of forward-backward stochastic differential equations (FBSDEs) with constraints. We use the penalization method to construct a sequence of FBSDEs without constraints, and we show that the solutions of these equations converge to the minimal solution we are interested in.

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Tian, Ran, and Zhiyong Yu. "Mean-field type FBSDEs in a domination-monotonicity framework and LQ multi-level Stackelberg games." Probability, Uncertainty and Quantitative Risk 7, no. 3 (2022): 215. http://dx.doi.org/10.3934/puqr.2022014.

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<p style='text-indent:20px;'>Motivated by various mean-field type linear-quadratic (MF-LQ, for short) multi-level Stackelberg games, we propose a kind of multi-level self-similar randomized domination-monotonicity structures. When the coefficients of a class of mean-field type forward-backward stochastic differential equations (MF-FBSDEs, for short) satisfy this kind of structures, we prove the existence, the uniqueness, an estimate and the continuous dependence on the coefficients of solutions. Further, the theoretical results are applied to construct unique Stackelberg equilibria for forward and backward MF-LQ multi-level Stackelberg games, respectively.</p>

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Zhao, Weidong, Wei Zhang, and Lili Ju. "A Numerical Method and its Error Estimates for the Decoupled Forward-Backward Stochastic Differential Equations." Communications in Computational Physics 15, no. 3 (March 2014): 618–46. http://dx.doi.org/10.4208/cicp.280113.190813a.

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AbstractIn this paper, a new numerical method for solving the decoupled forward-backward stochastic differential equations (FBSDEs) is proposed based on some specially derived reference equations. We rigorously analyze errors of the proposed method under general situations. Then we present error estimates for each of the specific cases when some classical numerical schemes for solving the forward SDE are taken in the method; in particular, we prove that the proposed method is second-order accurate if used together with the order-2.0 weak Taylor scheme for the SDE. Some examples are also given to numerically demonstrate the accuracy of the proposed method and verify the theoretical results.

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Zhao, Weidong, Tao Zhou, and Tao Kong. "High Order Numerical Schemes for Second-Order FBSDEs with Applications to Stochastic Optimal Control." Communications in Computational Physics 21, no. 3 (February 7, 2017): 808–34. http://dx.doi.org/10.4208/cicp.oa-2016-0056.

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AbstractThis is one of our series papers on multistep schemes for solving forward backward stochastic differential equations (FBSDEs) and related problems. Here we extend (with non-trivial updates) our multistep schemes in [W. Zhao, Y. Fu and T. Zhou, SIAM J. Sci. Comput., 36 (2014), pp. A1731-A1751] to solve the second-order FBSDEs (2FBSDEs). The key feature of the multistep schemes is that the Euler method is used to discretize the forward SDE, which dramatically reduces the entire computational complexity. Moreover, it is shown that the usual quantities of interest (e.g., the solution tuple (Yt,Zt,At,Γt) of the 2FBSDEs) are still of high order accuracy. Several numerical examples are given to show the effectiveness of the proposed numerical schemes. Applications of our numerical schemes to stochastic optimal control problems are also presented.

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Di Persio, Luca, Emanuele Lavagnoli, and Marco Patacca. "Calibrating FBSDEs Driven Models in Finance via NNs." Risks 10, no. 12 (November 30, 2022): 227. http://dx.doi.org/10.3390/risks10120227.

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The curse of dimensionality problem refers to a set of troubles arising when dealing with huge amount of data as happens, e.g., applying standard numerical methods to solve partial differential equations related to financial modeling. To overcome the latter issue, we propose a Deep Learning approach to efficiently approximate nonlinear functions characterizing financial models in a high dimension. In particular, we consider solving the Black–Scholes–Barenblatt non-linear stochastic differential equation via a forward-backward neural network, also calibrating the related stochastic volatility model when dealing with European options. The obtained results exhibit accurate approximations of the implied volatility surface. Specifically, our method seems to significantly reduce the neural network’s training time and the approximation error on the test set.

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Li, Min, and Zhen Wu. "Near-optimal control problems for forward-backward regime-switching systems." ESAIM: Control, Optimisation and Calculus of Variations 26 (2020): 94. http://dx.doi.org/10.1051/cocv/2020016.

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This paper investigates the near-optimality for a class of forward-backward stochastic differential equations (FBSDEs) with continuous-time finite state Markov chains. The control domains are not necessarily convex and the control variables do not enter forward diffusion term. Some new estimates for state and adjoint processes arise naturally when we consider the near-optimal control problem in the framework of regime-switching. Inspired by Ekeland’s variational principle and a spike variational technique, the necessary conditions are derived, which imply the near-minimum condition of the Hamiltonian function in an integral sense. Meanwhile, some certain convexity conditions and the near-minimum condition are sufficient for the near-optimal controls with order ε1/2. A recursive utility investment consumption problem is discussed to illustrate the effectiveness of our theoretical results.

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Huang, Hong, Xiangrong Wang, and Ying Li. "A Necessary Condition for Optimal Control of Forward-Backward Stochastic Control System with Lévy Process in Nonconvex Control Domain Case." Mathematical Problems in Engineering 2020 (June 3, 2020): 1–11. http://dx.doi.org/10.1155/2020/1768507.

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This paper analyzes one kind of optimal control problem which is described by forward-backward stochastic differential equations with Lévy process (FBSDEL). We derive a necessary condition for the existence of the optimal control by means of spike variational technique, while the control domain is not necessarily convex. Simultaneously, we also get the maximum principle for this control system when there are some initial and terminal state constraints. Finally, a financial example is discussed to illustrate the application of our result.

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Bai, Yu, Di Zhou, and Zhen He. "A Class of Pursuit Problems in 3D Space via Noncooperative Stochastic Differential Games." Aerospace 12, no. 1 (January 13, 2025): 50. https://doi.org/10.3390/aerospace12010050.

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This paper investigates three-dimensional pursuit problems in noncooperative stochastic differential games. By introducing a novel polynomial value function capable of addressing high-dimensional dynamic systems, the forward–backward stochastic differential equations (FBSDEs) for optimal strategies are derived. The uniqueness of the value function under bounded control inputs is rigorously established as a theoretical foundation. The proposed methodology constructs optimal closed-loop feedback strategies for both pursuers and evaders, ensuring state convergence and solution uniqueness. Furthermore, the Lebesgue measure of the barrier surface is computed, enabling the design of strategies for scenarios involving multiple pursuers and evaders. To validate its applicability, the method is applied to missile interception games. Simulations confirm that the optimal strategies enable pursuers to consistently intercept evaders under stochastic dynamics, demonstrating the robustness and practical relevance of the approach in pursuit–evasion problems.

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Xie, Tinghan, Bing-Chang Wang, and Jianhui Huang. "Robust linear quadratic mean field social control: A direct approach." ESAIM: Control, Optimisation and Calculus of Variations 27 (2021): 20. http://dx.doi.org/10.1051/cocv/2021021.

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This paper investigates a robust linear quadratic mean field team control problem. The model involves a global uncertainty drift which is common for a large number of weakly-coupled interactive agents. All agents treat the uncertainty as an adversarial agent to obtain a “worst case” disturbance. The direct approach is applied to solve the robust social control problem, where the state weight is allowed to be indefinite. Using variational analysis, we first obtain a set of forward-backward stochastic differential equations (FBSDEs) and the centralized controls which contain the population state average. Then the decentralized feedback-type controls are designed by mean field heuristics. Finally, the relevant asymptotically social optimality is further proved under proper conditions.

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Angiuli, Andrea, Christy V. Graves, Houzhi Li, Jean-François Chassagneux, François Delarue, and René Carmona. "Cemracs 2017: numerical probabilistic approach to MFG." ESAIM: Proceedings and Surveys 65 (2019): 84–113. http://dx.doi.org/10.1051/proc/201965084.

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This project investigates numerical methods for solving fully coupled forward-backward stochastic differential equations (FBSDEs) of McKean-Vlasov type. Having numerical solvers for such mean field FBSDEs is of interest because of the potential application of these equations to optimization problems over a large population, say for instance mean field games (MFG) and optimal mean field control problems. Theory for this kind of problems has met with great success since the early works on mean field games by Lasry and Lions, see [29], and by Huang, Caines, and Malhamé, see [26]. Generally speaking, the purpose is to understand the continuum limit of optimizers or of equilibria (say in Nash sense) as the number of underlying players tends to infinity. When approached from the probabilistic viewpoint, solutions to these control problems (or games) can be described by coupled mean field FBSDEs, meaning that the coefficients depend upon the own marginal laws of the solution. In this note, we detail two methods for solving such FBSDEs which we implement and apply to five benchmark problems. The first method uses a tree structure to represent the pathwise laws of the solution, whereas the second method uses a grid discretization to represent the time marginal laws of the solutions. Both are based on a Picard scheme; importantly, we combine each of them with a generic continuation method that permits to extend the time horizon (or equivalently the coupling strength between the two equations) for which the Picard iteration converges.

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43

Antonelli, Fabio. "Backward-Forward Stochastic Differential Equations." Annals of Applied Probability 3, no. 3 (August 1993): 777–93. http://dx.doi.org/10.1214/aoap/1177005363.

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Zhang, Qi. "Terminal-Dependent Statistical Inference for the Integral Form of FBSDE." Discrete Dynamics in Nature and Society 2013 (2013): 1–13. http://dx.doi.org/10.1155/2013/753025.

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Backward Stochastic Differential Equation (BSDE) has been well studied and widely applied. The main difference from the Original Stochastic Differential Equation (OSDE) is that the BSDE is designed to depend on a terminal condition, which is a key factor in some financial and ecological circumstances. However, to the best of knowledge, the terminal-dependent statistical inference for such a model has not been explored in the existing literature. This paper is concerned with the statistical inference for the integral form of Forward-Backward Stochastic Differential Equation (FBSDE). The reason why I use its integral form rather than the differential form is that the newly proposed inference procedure inherits the terminal-dependent characteristic. In this paper the FBSDE is first rewritten as a regression version, and then a semiparametric estimation procedure is proposed. Because of the integral form, the newly proposed regression version is more complex than the classical one, and thus the inference methods are somewhat different from those designed for the OSDE. Even so, the statistical properties of the new method are similar to the classical ones. Simulations are conducted to demonstrate finite sample behaviors of the proposed estimators.

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45

Yong, Jiongmin. "Linear ForwardBackward Stochastic Differential Equations." Applied Mathematics and Optimization 39, no. 1 (January 2, 1999): 93–119. http://dx.doi.org/10.1007/s002459900100.

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Rotenstein, Eduard. "A multi-dimensional FBSDE with quadratic generator and its applications." Analele Universitatii "Ovidius" Constanta - Seria Matematica 23, no. 2 (June 1, 2015): 213–22. http://dx.doi.org/10.1515/auom-2015-0038.

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Abstract We consider, in the Markovian framework, a multi-dimensional forward - back - ward stochastic differential equation with quadratic growth for the generator function of the backward system. We prove an existence result of the solution and we use this result for pricing and hedging of contingent claims that depend on non-tradeable indexes by portfolios consisting in correlated risky assets.

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Du, Kai, and Qi Zhang. "Semi-linear degenerate backward stochastic partial differential equations and associated forward–backward stochastic differential equations." Stochastic Processes and their Applications 123, no. 5 (May 2013): 1616–37. http://dx.doi.org/10.1016/j.spa.2013.01.005.

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Zhu, QingFeng, and YuFeng Shi. "Forward-backward doubly stochastic differential equations and related stochastic partial differential equations." Science China Mathematics 55, no. 12 (May 20, 2012): 2517–34. http://dx.doi.org/10.1007/s11425-012-4411-1.

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Peng, Shige, and Yufeng Shi. "Infinite horizon forward–backward stochastic differential equations." Stochastic Processes and their Applications 85, no. 1 (January 2000): 75–92. http://dx.doi.org/10.1016/s0304-4149(99)00066-6.

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Hu, Y., and S. Peng. "Solution of forward-backward stochastic differential equations." Probability Theory and Related Fields 103, no. 2 (June 1995): 273–83. http://dx.doi.org/10.1007/bf01204218.

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